Phonon Energies in Germanium from Phonon-Assisted Tunneling

Abstract

The line shapes of the peaks in the bias dependence of the second derivative of the phonon-assisted tunnel current with respect to voltage were measured for Sb-doped germanium tunnel junctions between 4.2 and 1.15\ifmmode^\circ\else\textdegree\fi{}K. The antimony concentration was 5\ifmmode\times\else\texttimes\fi{}${10}^{18}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. The phonon energies at the point $L$ in the Brillouin zone, determined to an accuracy of \ifmmode\pm\else\textpm\fi{}0.1%, are $E(\mathrm{TA})=7.766$, $E(\mathrm{LA})=27.58$, $E(\mathrm{LO})=30.62$, and $E(\mathrm{TO})=36.15$ in meV. This accuracy verifies the assumptions made in determining the pressure dependence of these phonon energies previously reported by the author. The major contribution to the line shapes is $\mathrm{kT}$ broadening. Minor contributions arise from a small bias dependence of the tunnel current per energy interval and from the phonon dispersion near $L$. Approximate values for the curvatures of the phonon spectrum at $L$ were determined. A lower limit of 1.6\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}11}$ sec was found for the lifetimes of these phonons. From double phonon emission processes the energy of the optic phonon at $\ensuremath{\Gamma}$ was found to be 37.3\ifmmode\pm\else\textpm\fi{}0.2 meV.